Notation for Bijection
$\Begingroup$ I Would Like to Specify an Explicit Bijection Between Small, Finite Sets. for Example, to Specify a Specific $F \Colon \{ A, B, C \} \To \{ X, Y...
I would like to specify an explicit bijection between small, finite sets.
For example, to specify a specific $f \colon \{ a,b,c \} \to \{ x,y,z \}$, I could define $f$ by
$$ f(a)=y \\ f(b)=x \\ f(c)=z $$
Is there some easily-recognizable notation for this, that is shorter than what I have written?
For example, I could write $f= \{ (a,y), (b,x), (c,z) \}$ or say that $f$ is given by $a \leftrightarrow y, b \leftrightarrow x$ and $c \leftrightarrow z$. Are there other (better) options?
3 Answers
There isn't a universally accepted notation for this. Any of the three options you mentioned should be fine. Though, for the third one, it might be better to replace "$\leftrightarrow$" with "$\mapsto$".
One suggestion is to view such bijections as permutations of the set $\{1,2,3\}$. And for those we have compact notations like cycle decomposition. Formally, let $X=\{x,y,z\}$ and $A=\{a,b,c\}$. Fix some bijections $\chi\colon \{1,2,3\}\to X$ and $\alpha\colon\{1,2,3\}\to A$. Then any bijection $X\to A$ is uniquely determined by some permutation $\sigma\in S_3$. That is, given $\sigma\in S_3$ we get a bijection $\alpha\circ\sigma\circ\chi^{-1}\colon X\to A$ and any bijection can be produced in such a way. Now take your favourite notation for peremutations.
You can write your function as $$ f= \begin{pmatrix} a & b & c \\ y & x & z \end{pmatrix} $$
If the domain is fixed (which is common), then the second line determines the function; thus, you may also simply write $ f=(y~x~z)$.
Since $z$ is fixed, you can also write $f=(y~x)$ with the convention that any element that does not appear is mapped to itself.
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